The geometry of the universal Teichmüller space and the Euler-Weil-Petersson equation

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The geometry of the universal Teichmüller space and the Euler-Weil-Petersson equation

Francois Gay-Balmaz, Tudor Ratiu

To cite this version:

Francois Gay-Balmaz, Tudor Ratiu. The geometry of the universal Teichmüller space and the Euler-Weil-Petersson equation. Advances in Mathematics, Elsevier, 2015, 279, pp.717-778.

�10.1016/j.aim.2015.04.005�. �hal-01396618�

Contents lists available atScienceDirect

Advances in Mathematics

www.elsevier.com/locate/aim

The geometry of the universal Teichmüller space and the Euler–Weil–Petersson equation

François Gay-Balmaz^a,∗,1,Tudor S. Ratiu^b,2

aCNRS-LMD-IPSL,EcoleNormaleSupérieuredeParis,Paris,France

bSectiondeMathématiques,EcolePolytechniqueFédéraledeLausanne, CH-1015 Lausanne,Switzerland

a r t i c l e i n f o a b s t r a c t

Fondlydedicatedtothememoryof ourfriendJerroldE.Marsden

Keywords:

UniversalTeichmüllerspace Weil–Peterssonmetric CriticalSobolevindexGeodesic Globalexistence

Regularity

OntheidentitycomponentoftheuniversalTeichmüllerspace endowedwith theTakhtajan–Teotopology,thegeodesicsof the Weil–Petersson metric are shown to exist for all time.

Thiscomponentisnaturallyasubgroupofthequasisymmetric homeomorphismsofthecircle.Viewedthisway,theregularity of its elements is shown to be H^32−ε for all ε > 0. The evolutionaryPDEassociatedtothespatialrepresentationof thegeodesics of theWeil–Petersson metric isderived using multiplication and composition below the critical Sobolev index3/2.Geodesiccompletenessisusedtointroducespecial classes of solutionsof thisPDE analogousto peakons.Our settingisusedto prove that there existsa unique geodesic betweeneach twoshapesintheplaneinthecontextof the applicationoftheWeil–Peterssonmetricinimaging.

* Correspondingauthor.

E-mailaddresses:francois.gay-balmaz@lmd.ens.fr(F. Gay-Balmaz),tudor.ratiu@epfl.ch(T.S. Ratiu).

1 Research partially supported by a Swiss NSF postdoctoral fellowship, by a “Projet Incitatif de Recherche”contractfromtheEcoleNormaleSupérieuredeParis,andbytheANRprojectGEOMFLUID (ANR-14-CE23-0002-01).

2 ResearchpartiallysupportedbyNCCRSwissMAPandgrant200021-140238,bothoftheSwissNational ScienceFoundation,aswellasbythegovernmentgrantoftheRussianFederationforsupportofresearch projects implemented by leading scientists, Lomonosov Moscow State University under the agreement No.11.G34.31.0054.

Completeness Patternrecognition

Contents

1. Introduction . . . . 718

2. TheuniversalTeichmüllerspace . . . . 720

3. Takhtajan–Teotheory . . . . 734

4. Sobolevmultiplicationandcompositionbelowcriticalindex . . . . 739

4.1. Multiplication . . . . 739

4.2. Composition . . . . 742

5. RegularityinTandQS(S¹)^H_◦ . . . . 755

6. CompletenessoftheuniversalTeichmüllerspace . . . . 761

7. TheEuler–Weil–Peterssonequation . . . . 763

8. TeichonsasparticularWPgeodesics . . . . 771

9. Applicationtopatternrecognition . . . . 773

Acknowledgments . . . . 776

References . . . . 776

1. Introduction

Thispaperestablishesalinkbetweenthreedistinctsubjects:conservativeevolutionary PDEs having a form similar to those appearing in fluid dynamics, the theory of the universalTeichmüllerspace, andthestudyofmapsat criticalSobolevindex.

ItiswellknownsincetheworkofArnold[4]thatthesolutionsoftheEulerequations are the spatial representation of the geodesics on the group of volume preserving dif- feomorphisms.Inmaterialrepresentation, theevolutionisgovernedbyasmoothvector field,thegeodesicspray,onthetangentbundleofthisdiffeomorphismgroup(see[10,6]).

This point of view not only leads to an elegant proof of well posedness but to many other results regarding the Euler equations. This is a remarkable property thatholds only insomespecificsituationssuch as theEuler equationsfor aperfectfluid [10], the incompressiblenon-homogeneousEulerequations[26],theaveragedEulerequations[27, 36],and the n-dimensionalCamassa–Holmequations [16]. Evenequations thatexhibit stronggeometric properties,suchas KdV,ingeneral,donothavethisproperty.

The universalTeichmüller space appearsinmany areas of mathematics and mathe- matical physics.For example,itisaspecialcoadjoint orbitoftheBott–Virasorogroup [30] and plays an important role in the theory of Riemann surfaces, several complex variables, andquasiconformalmaps[14,24,29].

The theoryof thegroupsof diffeomorphisms of ann-dimensional manifold endowed withaSobolevmanifoldstructure requiresdifferentiabilityclassstrictlyabove ⁿ₂+ 1.It is not even clear how to define agroup of diffeomorphisms at this critical index.This is reflected in the fact that a particle path flow associated to the Euler equations in dimensionatleast3,definedateverypointinthereferenceconfiguration,isknownonly fordifferentiability classstrictlybiggerthanthiscriticalindex.

In this paper we shall establish a connection between these three problems in the context of Weil–Petersson geometry on the universal Teichmüller space. The classical theory endows the universalTeichmüller space with agroup structure and an infinite dimensionalcomplex Banach manifold structure relative to which the inclusion of the TeichmüllerspacesofRiemannsurfacesisholomorphic.However,itisnotatopological groupand theformula forthe Weil–Petersson metricproposedin[30] is divergentand thusdoesnotdefine aRiemannianmetric.Thesebasicproblems wereovercome in[37]

whoendowedtheuniversalTeichmüllerspacewithadifferentcomplexHilbertmanifold structure in which the formula for the Weil–Petersson metric not only converges, but defines astrong metric, that is, ametric which induces the Hilbertspace topology on eachtangentspaces.TheyalsoshowthattheidentitycomponentofuniversalTeichmüller space is a topological group. With this manifold structure the tangent space at the identity is the space of functions on the circle of class H^3/2. Therefore, the identity componentofuniversalTeichmüllerspacetakes theplace ofdiffeomorphisms ofcritical SobolevclassH^3/2.

Weshallstudy thisgroupfromthepointofviewofmanifoldsofmaps byidentifying itwithasubgroupofthequasisymmetrichomeomorphismsofthecircle.Weshallprove thatallelementsofthisgroupareofclassH^32−εforallε>0.Thenweshallusethefact thatthe metricis strongto show that allgeodesics ofthe Weil–Petersson metricexist foralltime,thatis,wehavegeodesiccompleteness.Weshallalsoprovethatthisspaceis Cauchycomplete(somethingnotgenerallyimplied bygeodesiccompletenessininfinite dimensions)relativetothedistancefunctiondefinedbytheWeil–Peterssonmetric.

Thespatial formulationofthegeodesicequations turnsoutto be considerablymore involvedthaninthecaseoftheEulerequations.Weobtainanewequation,thatwecall the Euler–Weil–Petersson equation, and we show its solutions are C⁰ in H^3/2 and C¹ inH^1/2.AcomparisonofthetechnicaldifficultiesencounteredinthestudyoftheEuler equationsandoftheEuler–Weil–Peterssonequationisinorder.FortheEulerequations, themain technicalissueis theproofofthe smoothnessofthegeodesicspray, whilethe passage from material to spatial representation does notessentially require additional functionalanalytic developments.FortheEuler–Weil–Petersson equation,however,the situationis theopposite. Thesmoothnessspray follows directlyfrom the factthatthe metricisstrong,whereasthepassagefromthematerialtospatialrepresentationismuch more involved since it requires the composition and multiplication under the critical Sobolev exponent 3/2. We close the paper with two applications. The first one is the proof of long time existence of special solutions called Teichons by analogy with the peakonsfortheCamassa–Holmequations.Itturnsoutthatthese singularsolutionsare actuallysmootherthanthegenericgeodesics.Inthesecondapplication,weuseagainlong time existence of Weil–Petersson geodesics to positivelyaddress acomment of Sharon andMumford[35],namelythatthereexistsauniquegeodesicbetweeneachtwoshapes intheplane.

Planofthepaper.Section2reviewsthebasicfactsconcerningtheuniversalTeichmüller space T(1) endowed with its classical infinite dimensional complex Banach manifold

structure.Inparticular,werecallthatthismanifoldstructureisnotcompatiblewiththe naturalgroupoperationandthattheformulafortheWeil–PeterssonRiemannianmetric onT(1) isdivergent.ThesedifficultiesaresolvedbyendowingT(1) withanewcomplex Hilbert manifold structure, the Takhtajan–Teo topology, thatwe review in Section 3.

This approach allows us to define a new topological group and Hilbert manifold of homeomorphisms of the circle, thatreplaces the group of Sobolev diffeomorphisms in the caseof thecritical exponents = 3/2. Inparticular, we show thatthemodelspace is givenbySobolevH^3/2 vectorfields onthecircle. Section4isdevotedtotheproofof a technicalresult, namely, the compositionof diffeomorphisms of the circle of Sobolev class strictly lower than the critical index 3/2 is also a diffeomorphism of the same type. In Section5 we prove that this new Hilbert manifold is continuously embedded in the topological groupof all homeomorphismsof the circle that are of Sobolevclass H^3/2−ε for all ε > 0. In Section 6 we exploit the strongness of the Weil–Petersson metric to show that the Hilbert manifold T(1) is geodesically and Cauchy complete.

The passage from theLagrangianto thespatial formulation ofgeodesics iscarried out in Section7, using multiplication and compositionunder criticalexponents inSobolev spaces. Section 8 is concerned with particular solutions of the Euler–Weil–Petersson equation,calledTeichons,byanalogywiththepeakonsoftheCamassa–Holmequations.

These Teichonsare shownto be particularWeil–Petersson geodesics.Finally,Section9 considers applicationtoimagingfromthefunctional analyticpointofviewdevelopedin thepaper.

2. Theuniversal Teichmüllerspace

In this sectionwe recallsomebasic classicalfactsweshall need aboutuniversalTe- ichmüllerspace,forthereader’sconvenienceandtoestablishnotation.Amorecomplete expositioncanbefoundin[1,12,14,24,29].

Notation and some important facts. Let C be the Riemannsphere. Let the open unit disk in the complex plane be denoted by D := {z ∈ C | |z| < 1} and its exterior by D^∗ :={z∈C||z|>1}.Wedenotebyd²z theusualtwodimensionalLebesguemeasure onC,thatis,d²z:= ₂ⁱ(dz∧d¯z).

Consider the separable complex Banachspace L¹(D^∗) of integrable complexvalued functionsonD^∗.Itsdualcanbeisometricallyidentifiedwiththenon-separable complex BanachspaceL^∞(D^∗) ofessentiallyboundedcomplexvaluedfunctionsonD^∗.Inthecon- text ofTeichmüllertheory, theelementsofL^∞(D^∗) arecalledtheBeltrami differentials on D^∗.

Define theclosedsubspace A₁(D^∗) =

φ∈L¹(D^∗)φis holomorphic onD^∗

of L¹(D^∗). Its dual canbe identified with the quotient Banach space L^∞(D^∗)/N(D^∗), where

N(D^∗) :=

⎧⎨

⎩μ∈L^∞(D^∗) ˆ

D^∗

μ(z)φ(z)d²z= 0,∀φ∈A₁(D^∗)

⎫⎬

⎭

isthespaceofinfinitesimallytrivialBeltramidifferentials.Thereisacanonicalsplitting L^∞(D^∗) =N(D^∗)⊕Ω^−1,1(D^∗), (2.1) whereΩ^−1,1(D^∗) istheclosednon-separable BanachsubspaceofL^∞(D^∗) definedby

Ω^−1,1(D^∗) :=

μ∈L^∞(D^∗)μ(z) = (1− |z|²)²φ(z), φa holomorphic map onD^∗ . This projection allows us to identify L^∞(D^∗)/N(D^∗) with Ω^−1,1(D^∗), whose elements arecalled,bydefinition,harmonicBeltrami differentialson D^∗.Wecanwritethisspace as

Ω^−1,1(D^∗) :=

μ(z) = (1− |z|²)²φ(z)φ∈A_∞(D^∗) ,

whereA_∞(D^∗) isthenon-separable complexBanachspace A_∞(D^∗) =

φ holomorphic inD^∗ sup

z∈D^∗|φ(z)(1− |z|²)²|<∞

.

Alltheresultsofthissectionremainvalid whenD^∗ isreplacedbyD.

TherealLiegroupPSU(1,1).RecallthatthebiholomorphicmapsoftheRiemannsphere C areoftheform

z→ az+b

cz+d where a, b, c, d∈C, ad−bc= 1.

The set of such maps form a group under composition that is readily checked to be isomorphictothecomplexmatrixLiegroup

PSL(2,C) := SL(2,C)/{±I}

of complex dimension3, called the group of Möbius transformations. The subgroup of allbiholomorphicmapsoftheRiemannspherewhichpreservetheunitdiscDareofthe form

z→ az+b

¯bz+ ¯a, where a, b∈C, and |a|²− |b|²= 1.

ThisgroupisisomorphictotherealthreedimensionalLiegroup PSU(1,1) := SU(1,1)/{±I}.

The elements of PSU(1,1) preservethe unitcircle S¹ and are uniquelydetermined by theirrestrictiontothecircleS¹.Beingconformal,thesemapsareorientationpreserving on the disk. Since the circle inherits the natural boundary orientation,it follows that PSU(1,1) can be regarded as a subgroup of Diff₊(S¹), theorientation preserving C^∞ diffeomorphisms ofthecircleS¹.

View thisway, andusingthe standardchartθ→e^iθ ofthecircle, theLiealgebraof PSU(1,1) consistsofperiodicfunctionsoftheform

psu(1,1) ={f_a,b,c(θ) =a+bsinθ+ccosθ|a, b, c∈R}.

TheLiealgebrabracketonthisspaceoffunctionsisminustheusualJacobi–Liebracket on vector fields. This Lie algebrabracket is given as follows: for Lie algebra elements f(θ)∂/∂θandg(θ)∂/∂θ,theirbracketis

[f, g](θ) =g(θ)f(θ)−g(θ)f(θ).

Quasiconformal maps.Let φ:A→φ(A) beanorientation preserving homeomorphism definedonanopensubsetAofthecomplexplane.Themapφissaidtobequasiconformal ifit hasalldirectionalderivatives(inthesense ofdistributions)inL¹_loc(A) andifthere is μ∈L^∞(A) with μ _∞<1 suchthat

∂z¯φ=μ∂zφ. (2.2)

This is called the Beltrami equation with coefficientμ. If Aand φ(A) have boundaries which areJordancurves(that is,curveshomeomorphicto acircle),then anyquasicon- formal map on A extendsto an orientation preserving homeomorphism from cl(A) to cl(φ(A)) (seeTheoremI.8.2in[25]).

Inasimilarway,anorientationpreservinghomeomorphismbetweenRiemannsurfaces issaidtobequasiconformalifitslocalexpressionsarequasiconformalmapsbetweenopen subsetsofthecomplexplane.TheonlyRiemannsurfacewewillconsideristheRiemann sphereC.

TheuniversalTeichmüllerspace.Werecallbelowtwoequivalentmodelsfortheuniversal Teichmüller space, by following the presentation given in [37]. See also [1,24,29]. We denote byB₁^∗ theunitopenball inL^∞(D^∗).

•ModelA.Extendeveryμ∈B₁^∗to Dbythereflection

μ(z) =μ 1

z z²

z², z∈D.

Thus we get a new map,also denoted by μ∈ L^∞(C). Wedenote byω_μ :C → C the uniquesolutionoftheBeltramiequation

∂z¯ωμ=μ∂zωμ

whichfixes±1,−i.Thisω_μisobtainedbyapplyingtheexistenceanduniquenesstheorem ofAhlfors–Bers(see[2]);ω_μ isahomeomorphismofC anditsatisfies

ωμ(z) =ωμ

1 z

dueto thereflectionsymmetry ofμ.Asaresult,S¹,DandD^∗ areinvariantunderω_μ.

•ModelB. Extendeveryμ∈B^∗₁ to be zerooutsideD^∗. Wedenoteby ω^μ :C →C the uniquesolutionoftheBeltrami equation

∂z¯ω^μ=μ∂zω^μ,

satisfying the conditions f(0) = 0,∂zf(0) = 1, and ∂_z²f(0) = 0, where f is the holo- morphicmappingf :=ω^μ|D.This ω^μ isahomeomorphismof C andisalsoobtainedby applyingtheexistenceanduniquenesstheoremofAhlfors–Bers.

Therelationbetweenthesetwomodels isgivenbythefollowingstandardresult.

Theorem2.1. Forμ,ν ∈B₁^∗,wehave theequivalence ωμ|S¹ =ων|S¹ ⇐⇒ ω^μ|D=ω^ν|D.

See [24], Chapter III, Theorem 1.2 for aproof. We now recall the definition of the universalTeichmüller space.

Definition2.2.TheuniversalTeichmüller spaceisthequotientspace:

T(1) :=B₁^∗/∼, relativeto followingequivalencerelationonB₁^∗:

μ∼ν ⇐⇒ ωμ|S¹ =ων|S¹ ⇐⇒ ω^μ|D=ω^ν|D.

In viewof this definition T(1), endowed with the quotient topology, is clearly con- nected.ItturnsoutthatT(1)iscontractible([24],ChapterIII,Theorem 3.2).

The Bers embedding and the complex Banach manifold structure. The embedding of T(1) intoA_∞(D) playsacrucialinthetheoryofTeichmüllerspaces.Werecallbelowthe classicalBers theoremaboutthisembedding.

Theorem2.3. TheBers embedding

β :T(1)→A_∞(D), β([μ]) :=S(ω^μ|D),

isaninjectivemappingfromT(1)ontoanopensubsetofA_∞(D).Itsimagecontainsthe ball ofradius 2andiscontainedin theballof radius6.HereS denotes theSchwarzian derivative of aconformalmapf,that is,

S(f) := ∂³_zf

∂_zf −3 2

∂_z²f

∂_zf 2

.

Theorem2.4.ThereisauniqueBanachmanifoldstructureonT(1)relativetowhichthe projection map

π:B₁^∗→T(1)^B

is aholomorphic submersion.Relative tothis Banach manifoldstructure,theBers em- bedding

β :T(1)^B →A_∞(D) is abiholomorphicmaponto itsimage.

It can be shown that the kernel of the tangent map T0π : L^∞(D^∗) → T0T(1)^B is given byN(D^∗),so thetangentspaceT0T(1)^B canbe identifiedwiththeBanachspace L^∞(D^∗)/N(D^∗)Ω^−1,1(D^∗).

It isknownthatthetopologyofT(1) isthatofametricspacerelative totheTeich- müllerdistanceτ(see[24],ChapterIII,§2).Sincethisdistancefunctionwillnotbeused in the sequel, we will not recall the definition. Nevertheless, it is interesting to recall thatthe metricspace(T(1),τ) is complete.ByTheorem 2.3, T(1) ishomeomorphicto anopen subsetoftheBanachspaceA_∞(D),whichisclearlyincomplete.

Quasisymmetric homeomorphisms of the circle. An orientation preserving homeomor- phism η of thecircle S¹ isquasisymmetric ifthere isaconstantM such thatfor every xandevery|t|≤π/2

1

M ≤ η(x+t)−η(x) η(x)−η(x−t) ≤M.

Note thatthe definitionimpliesthatM ≥1.Here we identifythehomeomorphismsof the circle with the strictly increasing homeomorphisms of the real line satisfying the condition η(x+ 2π) = η(x)+ 2π. The set of all quasisymmetric homeomorphisms of the circleis denotedbyQS(S¹), itis agroupunderthecomposition ofmaps. Thelink withthequasiconformalmappingsonthediscisgivenbytheBeurling–Ahlforsextension theorem (see[5]).

Theorem 2.5. An orientation preserving homeomorphism of the circle admits a quasi- conformalextension tothediscif andonly ifitisquasisymmetric.

Notethatthisextensionisfarfrombeingunique.Fromthisresult,itfollowsthatthe restrictiontothecircleofasolutionωμ oftheBeltramiequationwithcoefficientμ∈B₁^∗ isaquasisymmetrichomeomorphismofthecircle.Wethereforeobtainthatthemap

Φ :T(1)−→QS(S¹)fix, [μ]−→ωμ|S¹ (2.3) isabijection, whereQS(S¹)_fix denotesthesubgroup ofQS(S¹) consistingof quasisym- metrichomeomorphismsfixingthepoints±1 and−i.

This bijection endows the group QS(S¹)fix with the structure of acomplex Banach manifoldbypushingforwardthisstructurefrom T(1)^B.TheresultingBanachmanifold isdenotedbyQS(S¹)^B_fix.ThisbijectionalsoendowsthesetT(1) withagroupstructure bypullingbackthegroupstructureofQS(S¹)_fix.A straightforwardcomputationshows thatthis groupstructure reads

[ν]·[μ] =

μ+ (ν◦ωμ)rμ

1 + ¯μ(ν◦ω_μ)r_μ

, rμ= ∂zωμ

∂_zω_μ. (2.4)

Relativeto the complex Banachmanifold structure, theright translationsR_[μ] are bi- holomorphic mappings for all [μ] ∈ T(1). The left translations are not continuous in general,thereforeT(1)^B isnotatopological group(seeTheorem 3.3in[24]).

NotethatQS(S¹)fix canbeidentified withthequotient spaceQS(S¹)/PSU(1,1) (or PSU(1,1)\QS(S¹)).Indeed,givenη∈QS(S¹),thereexistsonlyoneγ∈PSU(1,1) such thatη◦γ (or γ◦η) fixes thepoints ±1 and −i. Note thatthe projections QS(S¹) → QS(S¹)/PSU(1,1) and QS(S¹) → PSU(1,1)\QS(S¹) are not group homomorphisms, whenthequotientspaceisendowedwiththegroupstructureofQS(S¹)_fix.

Thetangent spaceofQS(S¹)^B_fix.RecallthatthetangentspacetoapointmofaBanach manifoldM is definedas a spaceof equivalenceof smoothcurves. Twocurvesare said to be equivalent at m if they are tangentat this point ina chart. In general there is notacanonical realization ofthe tangentspace. Nevertheless,inthe caseof manifolds ofmapssuchacanonicalrealization exists.

Werecallbelowhow tangentspaces tomanifoldsofmaps areconcretelyconstructed (see [32] and [10]). If s > dimM/2 then it is well-known that the set H^s(M,N) of H^s-Sobolevclassmapsbetweentwo boundarylesscompactmanifoldsM and N admits asmoothHilbert manifoldstructure. Letus recallthe basicideasof this construction.

Togetafeelingofwhatatangentvectoratf ∈H^s(M,N) mightbe,letustakeapath t ∈]−ε,ε[→ ft ∈ H^s(M,N) such that the map ft(m) is jointly smooth in (t,m) ∈ ]−ε,ε[×M. Then t ∈]−ε,ε[→ f_t(m) ∈ N is a smooth path in N and hence

∂f_t(m)/∂t|t=0isatangentvectortoN atthepointf₀(m).Thissuggeststhatatangent vectorat f isanH^s-map Uf :M →T N satisfyingUf(m)∈T_f(m)N foreverym∈M, thatis, avectorfieldcoveringf. Hencethecandidatetangentspaceis

TfH^s(M, N) ={Uf ∈H^s(M, T N)|Uf(m)∈T_f(m)N}.

NowoneproceedsconstructingchartsforH^s(M,N) withtheseHilbertspacesasmodels, usingtheexponentialmapofsomeRiemannianmetriconN.Oncethemanifoldstructure onH^s(M,N) hasbeenobtained,oneproves theidentity

d dtf_t

(m) = ∂

∂t(f_t(m)) forasmoothpatht∈]−ε,ε[→ft∈H^s(M,N).

Inourcase,QS(S¹)^B_fix isalsoaspaceofmaps,asopposedtoT(1)^B.Henceonewould like to study QS(S¹)^B_fix inthe spirit ofmanifolds of maps. However, thetopologiesare differentandtoimplementmanifoldofmapsconstructionsoneneedstousetheoremsin complex analysisas opposedto thestandardfactsinSobolevspacetheory.Our goalis to obtain aconcrete realization ofthe tangentspace at η:= Φ([μ]) ∈QS(S¹)^B_fix to the complexBanachmanifoldQS(S¹)^B_fix.Notethatwealreadyhaveanabstract description ofthistangentspace,namely,itisT_[μ]Φ

T_[μ]T(1)^B

.However,sofarwedonothaveany concrete realization of this complexBanachspace. Wewill show belowthatitis equal to theright translateof avery concrete function space on S¹, the Zygmund space. In theprocess wewillexplicitlycalculateT R_η.

Recall that theBanach manifold structure on QS(S¹)^B_fix is definedby the condition that the bijection Φ : T(1)^B → QS(S¹)fix, Φ([μ]) := ωμ|S¹ is a diffeomorphism. This simplysaysthatthemanifoldchartsofQS(S¹)^B_fixareoftheform(ϕ◦Φ⁻¹,Φ(U)) where (ϕ,U) arethemanifold chartsofT(1)^B.Acurveη_t∈QS(S¹)^B_fixis smoothifitisofthe form ηt =ω_μ(t)|S¹, where μ(t) isa smoothcurve inthe open ball B₁^∗. Theproblem of findingaconcreteexpressionofthevector _dt^dη_tisequivalenttothatoffindingaconcrete realization of the tangentspacesTηQS(S¹)fix or aconcrete expression forTηΦ. Afirst step inthis directionisthefollowing theorem.Thefirstpartis adirectconsequenceof Theorem 11 in[2]. Theexpression (2.5) is thereformulation forthe disk of Eq. (2.34) in [29]; see §1.2.11–1.2.12 ofthis book for additionalinformation and the proofof this formula.

Theorem 2.6.Letμ(t)∈B₁^∗ be asmoothcurvesuch thatμ(0)= 0.Thenforallz∈S¹, thecurvet→ω_μ(t)(z)∈S¹issmoothinaneighborhoodoft= 0.Thederivativeatt= 0 is givenby

∂

∂t

t=0

ωμ(t)(z) =Vν(z), where ν∈L^∞(C)isμ(0)˙ extendedtoCby reflection,and

Vν(z) =−(z+ 1)(z+i)(z−1) π

¨

C

ν(w)

(w+ 1)(w+i)(w−1)(w−z)d²w. (2.5)

Thistheoremgeneralizestothecasewhereμ(0) isnot0.Indeed,sincerighttranslation onB₁^∗issmooth,thecurvet→μ(t)·μ(0)⁻¹ issmooth.Herethedotdenotesthegroup multiplicationonB₁^∗ givenin(2.4). Wehave

∂

∂t

t=0

ω_μ(t)(z) = ∂

∂t

t=0

ω_μ(t)·μ(0)−1(ω_μ(0)(z)) =V_ν(ω_μ(0)(z)), whereν istheextension of _∂t^∂

t=0(μ(t)·μ(0)⁻¹) byreflection.

Hereisareformulationoftheseresults.Letηtbe asmoothcurveinQS(S¹)^B_fix. Then forallz∈S¹, thecurveηt(z) isdifferentiable asacurveonS¹ and thetimederivative isoftheform

∂

∂s

t=0

ηt(z) =Vν(η0(z)), (2.6)

whereVν isavectorfieldonS¹ oftheform (2.5).

ThenexttheoremstatesthatthevectorfieldVν belongstotheZygmundspaceonS¹ definedby

Z(S¹) :=

u∈C⁰(S¹) there is aCsuch that

|u(x+t) +u(x−t)−2u(x)| ≤C|t|for allx, t∈S¹ .

Here, the continuous vector fields u on the circle are identified with continuous 2π-periodicfunctionsonthereal line.Wealsodefinethesubspace

Z(S¹)0:=

u∈Z(S¹)|u(±1) =u(−i) = 0 .

Relativeto theZygmundnorm u Z:= u _∞+ sup

x,t∈S¹

|u(x+t) +u(x−t)−2u(x)|

|t| , (2.7)

Z(S¹) isanonseparableBanachspaceandZ(S¹)0 aclosedsubspace(see[9,14]).

Itisknownthatforall0< α <1 ands<1 wehavethestrictcontinuousinclusions Λ¹(S¹)⊂Z(S¹)⊂Λ^α(S¹), and Z(S¹)⊂H^s(S¹),

whereΛ¹(S¹) denotesthespaceofLipschitzfunctionsonthecircle,Λ^α(S¹) denotesthe spaceofα-Hölderfunctionsonthecircle,andH^s(S¹) denotesthespaceofSobolevclass H^sfunctionsonthecircle. IntermsoftheFourierseriesrepresentation wehave

H^s(S¹) =

u(x) =

n∈Z

u_ne^inx

u_−n =u_n and

n∈Z

|n|^2s|u_n|²<∞

.

These continuous inclusions are particular cases of embedding theorems for spaces of Besov–Triebel–Lizorkin type,see [39]and[34] forexample.

Theorem 2.7.Forallν ∈L^∞(D^∗),wehave Vν∈Z(S¹)0.Moreoverthelinearmap [ν]∈L^∞(D^∗)/N(D^∗)Ω^−1,1(D^∗)→Vν∈Z(S¹)0 (2.8) is an isomorphism of Banach spaces, where L^∞(D^∗)/N(D^∗) is endowed with the quo- tient norm andZ(S¹)0 is endowed with thecross-ratio norm, a normequivalent to the Zygmund normdefinedin (2.7).

SeeChapter3in[14],and[13,15]forthedefinitionofthecross-rationormandproofs of this theorem.OnZ(S¹) thecross-ratio normis actuallyaseminorm whose kernel is given bypsu(1,1).

Using the preceding discussion, it follows that for two smooth curves η_t¹ and η_t² in QS(S¹)^B_fix such thatη₀² = η₀¹ = η, theyare tangent at the point η with respect to the Banachmanifoldstructure ifandonlyif

∂

∂t(η¹_t(z)) = ∂

∂t(η_t²(z)), ∀z∈S¹.

By formula(2.6) and Theorem 2.7it follows thatarealization of thetangentspace to QS(S¹)^B_fix atη is

TηQS(S¹)^B_fix=Z(S¹)0◦η.

Remark. It is worth noting here the difference between theusual theory of H^s-diffeo- morphismgroupsandQS(S¹)^B_fix.TheformulaabovecompletelydeterminesTηQS(S¹)^B_fix. The sameformulais alsovalid forthe groupofH^s-diffeomorphisms Diff^s(M),namely, T_ηDiff^s(M)=X^s(M)◦η,where X^s(M) is theHilbertspace ofH^s-vectorfields onM.

However, forthediffeomorphismgrouponecangofurtherand saythatTηDiff^s(M)= X^s(M)◦η={Uη ∈H^s(M,T M)|Uη(m)∈T_η(m)M}, arealization thatisnotavailable forT_ηQS(S¹)^B_fix.

Withrespect tothisrealization,thetangentmapto Φ at[0] is

T_[0]Φ :L^∞(D^∗)/N(D^∗)→T_idQS_fix(S¹) =Z(S¹)₀, T_[0]Φ([ν]) =V_ν, and thederivativeofasmoothcurveηt isthevectorinTηtQS(S¹)^B_fix givenby

d dtηt

(z) = ∂

∂t(ηt(z)).

Of course,theprevious equalityholds onlyifthe curveηt is knownto be smooth with respecttotheBanachmanifoldstructureinducedbyΦ.Itisnotsufficientthatforallz, thecurvet→ηt(z) issmooth.

RighttranslationbyγisgivenbyRγ(ξ)=ξ◦γanditisknowntobeasmoothmap.

Usingtheprecedingresultswehave,foruη∈TηQS(S¹)^B_fix, (T Rγ(uη)) (z) =

d dt

t=0

Rγ(ηt)

(z) = d dt

t=0

∂

∂t(ηt(γ(z)))

= d

dt

t=0

ηt

(γ(z)) =uη(γ(z)).

Thus,wehaveT Rγ(uη)=uη◦γ.

The isomorphism between the tangent space TidQS(S¹)^B_fix and the Banach model space A_∞(D) of T(1)^B is given bytaking thetangent map at the identity to the map β◦Φ⁻¹: QS(S¹)^B_fix→T(1)^B→A_∞(D), whereβ denotestheBersembeddingand Φ is thediffeomorphismdefinedin(2.3).It wasproved[38],Theorem2.11,thatwehave

Tid

β◦Φ⁻¹

:

n∈Z

une^inx∈TidQS(S¹)^B_fix−→i

n≥2

(n³−n)unzⁿ⁻²∈A_∞(D).

UsingthecomplexBanachmanifoldstructureofQS(S¹)^B_fix thoughtofasareal man- ifold, it is possible to endow the whole group QS(S¹) with a real Banach manifold structure,bydeclaringthatthebijection

Ψ : QS(S¹)−→PSU(1,1)×QS(S¹)^B_fix, (2.9) definedbythecondition

Ψ(η) = (ˆη, η₀)⇐⇒η= ˆη◦η₀, (2.10) is adiffeomorphism. The groupQS(S¹) endowed with this Banachmanifold structure isdenoted byQS(S¹)^B.Its propertiesare giveninthetheorem below. Wewill usethe following lemma which shows thatthe choice of another subgroup fixing three points doesnotchangetheBanachmanifoldstructure onQS(S¹).

Lemma 2.8. Let QS(S¹)₁ be a subgroup of QS(S¹)consisting of quasisymmetric home- omorphisms fixing threepoints. Then QS(S¹)1 can be endowed witha Banachmanifold structurein thesame wayasQS(S¹)fix.The bijection

PSU(1,1)×QS(S¹)^B_fix→PSU(1,1)×QS(S¹)^B₁ definedby

(γ₀, η₀)→(γ₁, η₁), such that γ₀◦η₀=γ₁◦η₁, isasmoothdiffeomorphism.

Proof. Bydefinitionof the Banachmanifold structures onQS(S¹)fix and QS(S¹)1, we obtainthatthemap

QS(S¹)fix→QS(S¹)1, η →γ◦η, (2.11) where γistheuniqueMöbiustransformationsuchthatγ◦η∈QS(S¹)1,isadiffeomor- phism.

Wenow showthatthemap (γ₀,η₀)→(γ₁,η₁) issmooth. Sinceγ₀◦η₀ =γ₁◦η₁,we have η₁ = (γ₁⁻¹◦γ₂)◦η₀. Thus, using (2.11), we obtainthatthe map (γ₀,η₀)→η₁ is smooth. Inorder toshowthat(γ0,η0)→γ1is smooth,weconsiderthemap

F : PSU(1,1)×QS(S¹)fix×PSU(1,1)→R³, F(γ, η, ξ) = (γ(η(xi))−ξ(xi)), where (xi),i = 1,2,3, denote the fixed points associated to the group QS(S¹)1. Note thatF is smoothandthatF(γ0,η0,ξ)= 0 ifandonlyif ξ=γ1. Thepartial derivative ofF withrespecttothevariableξandinthedirectionV ∈TξPSU(1,1) iscomputedto be

∂F

∂ξ (γ0, η0, ξ)(V) =−(V(xi)),

therefore the linearmap ^∂F_∂ξ(γ0,η0,ξ) : TξPSU(1,1) → R³ is an isomorphism, and by theimplicitfunctiontheorem,thecorrespondence(γ0,η0)→η1 issmooth. 2

As a consequence of this lemma, we obtain that the identification of QS(S¹) with PSU(1,1)×QS(S¹)^B_fixorPSU(1,1)×QS(S¹)^B₁ givesthesameBanachmanifoldstructure.

Theorem2.9. Thetangent spaceattheidentity totherealBanachmanifoldQS(S¹)^B is the Zygmund space Z(S¹). The group QS(S¹)^B is not atopological group butthe right translations aresmooth; QS(S¹)^B containsthesubgroupQS(S¹)^B_fix asaclosed submani- fold ofcodimension3.

Proof. FromthedefinitionoftheBanachmanifoldstructure wehave TidQS(S¹)^B =psu(1,1)⊕TidQS(S¹)^B_fix.

RecallthatT_idQS(S¹)^B_fixconsistsofvectorfieldsinZ(S¹) suchthatu(±1)=u(−i)= 0.

Thereforebyaddinganyelementofpsu(1,1) werecoverthewholespaceZ(S¹).Theset QS(S¹)^B_fix is clearly a subgroup of QS(S¹)^B. It is also aclosed submanifoldsince it is identified withtheclosedsubmanifold{e}×QS(S¹)^B_fixinPSU(1,1)×QS(S¹)^B_fix.

We now show that the right translations R_ξ : QS(S¹)^B → QS(S¹)^B,η → η◦ξ are smooth for each fixed ξ ∈ QS(S¹). We first prove this for ξ ∈ QS(S¹)_fix. Using the diffeomorphism (2.9), the correspondence η → η◦ξ reads (ˆη,η0) → (ˆη,η0◦ξ). This

is a smooth map since right translations are known to be smooth on QS(S¹)^B_fix. We now consider thecase ξ ∈ PSU(1,1).Note thatfor any ξ we candefine the subgroup QS(S¹)ξ consisting of quasisymmetric homeomorphisms of the circle fixing the three pointsξ⁻¹(−1),ξ⁻¹(1) andξ⁻¹(−i).AsamapfromPSU(1,1)×QS(S¹)^B_fixtoPSU(1,1)× QS(S¹)^B_ξ ,thecorrespondenceη→η◦ξreads(ˆη,η₀)→(ˆη◦ξ,ξ⁻¹◦η₀◦ξ).Bythepreceding lemma it suffices to show that this last correspondence is smooth. For the first factor this is trivial sinceright translation onPSU(1,1) is smooth. Henceit suffices to show thatthemap

η0∈QS(S¹)^B_fix→ξ⁻¹◦η0◦ξ∈QS(S¹)^B_ξ

issmooth.Thisfollows fromthefactthis mapisinducedbythesmoothmap μ∈L^∞(D^∗)→(μ◦ξ)∂zξ

∂_zξ ∈L^∞(D^∗).

ToshowthatR_ξ isasmoothmapforallξ∈QS(S¹) itsufficestowriteξ= ˆξ◦ξ₀with ( ˆξ,ξ0)∈PSU(1,1)×QS(S¹)fix.WethenhaveRξ =Rξ0◦Rξˆ,whichisacompositionof smoothmapsbytheprecedingarguments. 2

Asin thecaseof QS(S¹)^B_fix, wecanshow thatTηQS(S¹)^B =Z(S¹)◦η. Letηt be a smoothcurveinQS(S¹)^B.Bydefinition,see(2.10),wecanwriteηt= ˆηt◦(ηt)0,where ˆ

ηt is asmoothcurve inPSU(1,1) and(ηt)0 isasmooth curve inQS(S¹)^B_fix. Therefore, weobtainthatforallz∈S¹ thecurveη_t(z) issmooth.Adirectcomputationshowsfor asmoothcurveηtwehave

∂

∂t

t=0

ηt(z) =V(η0(z)),

where V ∈ Z(S¹). This shows that a canonical realization of the tangent space TηQS(S¹)^B is given by Z(S¹)◦η, and that the tangent map to right translation is T R_γ(u_η)=u_η◦γ.

A system of neighborhoods of the identity inQS(S¹)^B is given by {U(ε) | ε > 0}, whereU(ε) consistsofallquasisymmetrichomeomorphismsη∈QS(S¹) suchthat

1

1 +ε ≤ η(x+t)−η(x)

η(x)−η(x−t) ≤1 +ε and sup

x∈S¹

|η(x)−x|,|η⁻¹(x)−x|

< ε.

Atotherpoints,theneighborhoodsareobtainedbyrighttranslation.

Relationwithdiffeomorphismgroups.Wehavethefollowingchainofsubgroupinclusions Diff+(S¹)⊂Diff^s₊(S¹)⊂Diff^C₊¹(S¹)⊂QS(S¹), (2.12)

for all s > 3/2. The differential properties are the following. The group Diff+(S¹) is endowed with the C^∞ Fréchet manifold structure. The group Diff^s₊(S¹) denotes the groupof allorientation preservingSobolev classH^sdiffeomorphisms of thecircle. Itis endowedwiththeSobolevH^sHilbertmanifoldstructure;thisispossibleforalls>3/2.

The group Diff^C₊¹(S¹) is endowed with the C¹ Banach manifold structure. All these manifold structures are real and not complex. Recall that Diff₊(S¹) is a Fréchet Lie group (see [21]), Diff^s₊(S¹) and Diff^C₊¹(S¹) are topological groups with smooth right translations[10,31],QS(S¹) hassmoothrighttranslationsbutisnotatopologicalgroup (Theorem 2.9). Note also that all the inclusions are smooth. The two first inclusions on the left have dense ranges and the last inclusion on the right is neither dense nor closed. The closure ofDiff^C₊¹(S¹) in QS(S¹) determines the topological groupS(S¹) of symmetric homeomorphismsofthecircle. Wereferto [15,14]forthedefinitionand the propertiesofS(S¹).

Thesamedifferentialpropertiesholdforthecorrespondingsubgroupsfixingthepoints

±1 and−i.Wegettheinclusions

Diff₊(S¹)_fix⊂Diff^s₊(S¹)_fix⊂Diff^C₊¹(S¹)_fix⊂QS(S¹)_fix, (2.13) foralls>3/2.Thesesubgroupshavetheadditionalproperty tobealsocomplexmani- folds.Thetangentspacesattheidentityto thesesubgroups,denotedby

C^∞(S¹)₀⊂H^s(S¹)₀⊂C¹(S¹)₀⊂Z(S¹)₀, (2.14) are obtained by imposing the conditions u(±1) = u(−i) = 0 on the elements of the tangentspacesattheidentity tothecorrespondinglargegroupsin(2.12).

Notethatanotherrealizationofthetangentspacesattheidentitytothesesubgroups is givenbyimposingtheconditionsu₋₁=u₀=u₁= 0 ontheFouriercoefficients.This correspondstothinkingofthesesubgroupsasquotientspacesofthecorrespondinggroups by the MöbiusgroupPSU(1,1);therefore the vectorfields are taken modulopsu(1,1).

Thetangentspacesattheidentity inthisinterpretationaredenotedby h^∞⊂h^s⊂h^C1 ⊂h^QS.

More on thecomplex structure.Recall thatthecomplexstructure ofT(1)^B isthe triv- ial oneinduced by the assumption that the projection B₁^∗ → T(1)^B is a holomorphic submersion.Thereforethecomplexstructure operatorissimplymultiplicationbyi.We denote byJ the complexstructureoperatorinducedontheBanachmanifoldQS(S¹)^B_fix. Thefollowingtheorem dueto[30]showsthatJ takesaremarkablysimpleexpressionin termsofFourierseries.

Theorem 2.10. The complex structure on the Banach manifold QS(S¹)^B_fix is the right- invariantstructure givenattheidentity bythemapJ :h^QS→h^QS definedby

J

⎛

⎝

n=−1,0,1

u_ne^inx

⎞

⎠=i

n=−1,0,1

sgn(n)u_ne^inx.

TheoperatorJ isinfacttheHilberttransformonthecircle J(u)(x) = 1

2π ˆ

S¹

u(s) cot s−x

2

ds.

For J,we keep the sign conventions in[37] which differ byan overall minussign from theonein[30].

TheWeil–Peterssonmetric.TheWeil–PeterssonHermitianmetriconT(1)^Bistheright- invariantmetricwhosevalueattheidentity[0] isgivenby

μ, ν:=

ˆ

D^∗

μ(z)ν(z) 4

(1− |z|²)²d²z. (2.15) This metric was introduced by Nag and Verjovsky [30] as a direct generalization of the Weil–Petersson metric on the finite dimensional Teichmüller spaces. As remarked by these authors,this Hermitian metric doesnot make sense for all μ,ν ∈ Ω^−1,1(D^∗).

Indeed,itconvergesonlyforμ,ν intheHilbertspaceH^−1,1(D^∗)⊂Ω^−1,1(D^∗) definedby

H^−1,1(D^∗) =

⎧⎨

⎩μ∈Ω^−1,1(D^∗) ˆ

D^∗

|μ(z)|² 1

(1− |z|²)²d²z <∞

⎫⎬

⎭

=

μ(z) = (1− |z|²)²φ(z)φ∈A2(D^∗)

,

where

A2(D^∗) =

⎧⎨

⎩φ holomorphic inD^∗ ˆ

D^∗

|φ(z)|²(1− |z|²)²d²z <∞

⎫⎬

⎭.

Usingtheidentifications

TidQS(S¹)^B_fix=h^QS

T_[0]Φ

←−−−− T_[0]T(1)^B

T_[0]β

−−−−→ A_∞(D), themetriconh^QS hastheexpression

hid(u, v) =π 2

∞ n=2

n(n²−1)unvn (2.16)